Bootstrap Initialization of MLE for Infinite Mixture Distributions with Applications in Insurance Data

Maximum likelihood estimation (MLE) in infinite mixture distributions often lacks closed-form solutions, requiring numerical methods such as the Newton–Raphson algorithm. Selecting appropriate initial values is a critical challenge in these procedures. This study introduces a bootstrap-based approach to determine initial parameter values for MLE, employing both nonparametric and parametric bootstrap methods to generate the mixing distribution. Monte Carlo simulations across multiple cases demonstrate that the bootstrap-based approaches, especially the nonparametric bootstrap, provide reliable and efficient initialization and yield consistent maximum likelihood estimates even when raw moments are undefined. The practical applicability of the method is illustrated using three empirical datasets: third-party liability claims in Indonesia, automobile insurance claim frequency in Australia, and total car accident costs in Spain. The results indicate stable convergence, accurate parameter estimation, and improved reliability for actuarial applications, including premium calculation and risk assessment. The proposed approach offers a robust and versatile tool both for research and in practice in complex or nonstandard mixture distributions.